Исследование локальной управляемости неголономной системы управления описывающей движение снейкборда.

ISSN 1810-0198. ??????? ???, ?. 20, ???. 5, 2015
??????? ???? ??????????, ???????? ??????????????? ???????????? ????????????????? ???????????, ?. ?????, ?????????? ?????????, ?????? ??????-?????????????? ????, ?????????, ????????? ??????? ?????????????? ?????? ? ?????????????? ??????? ? ?????????, e-mail:
[email protected]
Simonov Pyotr Mihkailovich, Perm State National Research University, Perm, the Russian Federation,
Doctor of Physical and Mathematical Sciences, Professor of the Information Systems and Mathematical
Methods in Economics Department, e-mail: [email protected]
??? 517.977.1
???????????? ????????? ????????????? ????????????
??????? ??????????, ??????????? ????????
??????????
c
?.?. ????????
???????? ?????: ?????????-????????????? ??????? ??????????; ????? ????????; ???????????? ???????; ?????????; ?????????????.
??????????? ?????? ?????????????? ???????? ????????? ????????????? ??? ???????????? ???????, ??????????? ???????? ??????????, ? ?????????????? ?????????-????????????? ??????? ??????????.
???????????? ?????? ?????????? ??? ?????? ???????????? ?????? ???????? ?????????
?????????????? ????????, ???????? ? ????? ?? ????????????? ????????? ?????????????.
???????????? ??????? ??? ????????????? ???????? ????????? ???????? ?????????? ??????? ? ???????-????????????? ??????????? ? ????? ?????? ??????????? ????????????? ????????? ?????????? [1]. ? ??????????? ??????? ???????????? ???????????, ? ?????
??????????? ??????? ????????????? ??? ???????????? ?????? ??????????? ? ????????
??????? ??????????????? ?????? ??, ??? ?? ????????? ????????????? ????????? ????????????? ?????????? ???????, ?. ?. ?? ?????????? ??????????? ?????? ????? ?????????????
?????? ?? ??? ???????? ???????????????? ????????? ???????. ? ?????? ?????? ???????????
???? ????? ???????????? ????????????? ???????????? ?????? ? ????? ???????? [2].
????? ?????? ??????? ??????????
x? = f (x, u),
x ? D ? Rn ,
u ? U ? Rm ,
(1)
??? x ? ??????? ??????, u ? ??????????; ????? ?????????? ??????????? ?? ??????? t .
????? ????????????, ??? D ? ???????, 0 ? D , U ? ????????? ???????, f ? C 1 (D О U ) .
????? ???????????? ????????? ??????????? ?????????????:
? ? ? ? ? ? ? ? ? ? ?. ??????? (1) ?????????? ???????????, ???? ??? ????? ?????
x(0) , x(1) ? D ???????? ? > 0, u ? L? ([0, ? ] ? U ) , ??? ??????? ??????? (1) ????? ???????
x(t) , t ? [0, ? ] , ??????????????? ????????? ???????? x(0) = x(0) , x(? ) = x(1) .
???? ? ??????????? ???????? ??????? ?????????? ??? ????? x(0) , x(1) ?? ?????????
? -??????????? ????? x = 0 , ?? ??????? (1) ?????????? ???????? ??????????? ? ?????
x = 0.
???????? ???????, ???????????? ? ?????? [3], ??????????? ??? ??????????? ????????????:
1018
ISSN 1810-0198. ??????? ???, ?. 20, ???. 5, 2015
? ? ? ? ? ? ? 1. ??? ?????????????? ? > 0 ?????????? ????????? ?????????? u(t, a) ?
? U , 0 6 t 6 ? , ????????? ?? ????????? a = (a1 , . . . , ak ) ? Rk . ???????????, ??? ???
?????? a ? Rk ?????????? ???????????? x(t, a) ??????? ?????? ????:
x?(t, a) = f x(t, a), u(t, a) , x ? D ? Rn , t ? [0, ? ];
x(0, a) = 0.
(2)
(3)
????? ?????????? a = a? , ??? a? ? Rk , ??? ??????? ??????????????? ???????
x(t, a? ) ????????????? ???????
x(?, a? ) = 0.
?????, ????
rank
?x(?, a)
?a
!
= n,
(4)
a=a?
?? ??????? (1) ???????? ?????????? ? ??????????? ????? x = 0 .
????????? ???????? ????????? ???????????? ????????? ????????? (?????????) ? ??????? ????? (????????). ??? ? ? ?????????? ? ?????????? ??????? ?????? ? ????????, ??????,
??????? ???????? ??????????? ? ?????????. ????????? ???????? ?????????, ?????? ???
????? ??????? ?? ???? ??????????? ??????. ????????? ???????? ???????? ? ???????? ???????? ???????? ? ?????? ?.?. ???????? [4].
?????????? ??????? ??????????
?
?
x? = V cos ?,
?
?
y? = V sin ?,
(5)
V sin ?
?
?
, V, ? ? [?1, +1].
? ?? =
l cos ?
??? x, y ? ?????????? ?????? ???? ???????, V ? ???????? ?????? ???? ??????????, l ?
?????????? ?? ?????? ???? ??????? ?? ????? ????????? ???? ?????, ? ? ???? ????????
????????, ? ? ???? ????? ?????????? ???? ?????????? ? ???? Ox (??. ???. 1).
???. 1: ?????? ??????????
1019
ISSN 1810-0198. ??????? ???, ?. 20, ???. 5, 2015
??????? (5) ???????? ??????? ???????? ?????????? ? ?????? ?????????? ????? ???????? ? ? ????????? ?????? ???? V .
??? k > 1, j = 2, 3, .h. . , k ? ??????? a = (a1 , . . . , ak ) , aj > 0 , ????????? ????? Sj
Pj?1 Pj
???????????? ???? Sj =
i=1 ai ,
i=1 ai , ? ?????? j = 1 ??????? S1 = [0, a1 ) .
????????? ??????? ?????????? u(t, a) ????????? ???????:
?
S[ k+1
?
2 ]
?
S2r?1 ;
? +1, ???? t ? r=1
k
S
[
]
(6)
u(t, a) =
2
?1, ???? t ?
S2r ;
?
?
? 0, ???? t > a +r=1
a + иии + a .
1
2
k
??????? k = 3 ? ?????????? ?????????? V = V (t, a) , ? = ?(t, a) ???? (6) , ???????,
??? ?????? ??????? (5) ???????? ?????????? ? ??????????? x = 0 ?? ??????? 1 ???
????????? ???????? ??????????:
?
? a1 = a1 ,
a2 = 4.03nl,
?
a3 = ?a1 + 4.03nl.
??? n, l ? Z , 0 < a1 < 4.03nl .
??????????
1. Laumond J.P. Robot Motion Planning and Control // Lecture Notes in Control and Information Sciences.
1998. ? 229. P. 343.
2. Coron J.-M. Return method: some applications to ?ow control // Mathematical control theory. Trieste:
The Abdus Salam ICTP. 2002. P. 655?704.
3. ???? ?.?., ????????? ?.?. ???????????? ????????? ????????????? ?????????? ?????? ???????
???????? // ???????? ???????? ????. 2008. ???. 38. ?. 136-144.
4. ??????? ?.?. ?????????????? ?????? ?????????? // ?????????????? ?????????????. 2006. ?. 18.
? 5. ?. 37-48.
????????? ? ???????? 11 ???? 2015 ?.
Astakhova T.N. INVESTIGATION OF LOCAL CONTROLLABILITY OF NONHOLONOMIC
CONTROL SYSTEM DESCRIBING THE SNAKEBOARD MOTION
The constructive test problem of local controllability of nonholonomic system describing the snakeboard motion by using sampled control function is considered.
Key words: sampled control function; return method; nonholonomic system; snakeboard; controllability.
???????? ??????? ??????????, ????????????? ??????????????? ?????????-????????????? ???????????, ?. ?????????, ?????????? ?????????, ???????? ??????-?????????????? ????, ?????? ??????? ?????????????? ?????? ? ??????????, e-mail: [email protected]
Astakhova Tatyana Nikolaevna, Nizhny Novgorod State Engineering and Economic University, Knyaginino, the Russian Federation, Candidate of Physics and Mathematics, Associate Professor of the Information Systems and Technology Department, e-mail: [email protected]
1020